SIAM CSE 2017 Loewner based water...
Transcript of SIAM CSE 2017 Loewner based water...
Tensor Based Approaches in Magnetic Resonance Spectroscopic Signal Analysis
Sabine Van Huffel, H. N. Bharath and Diana Sima
02/03/2017 – SIAM CSE17
Outline
APPLICATION 1: water suppression in Magnetic Resonance
spectroscopic imaging (MRSI)
� Method Æ Löwner based tensor approach applied to MRSI
APPLICATION 2: Tissue type differentiation of gliomas
� Method 1Æ Non-negative (N) CPD applied to MRSI
� Method 2Æ NCPD applied to multi-parametric MRI
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APPLICATION 1: water suppression in MRSI
𝑆 𝑡 = 𝑟=1
𝑅
𝑎𝑟𝑒𝑗𝜙𝑟𝑒(−𝑑𝑟+𝑗2𝜋𝑓𝑟) + 𝜂(𝑡)
Time domain Model𝑆 𝑓 = 𝑟=1
𝑅𝑎𝑟𝑒𝑗𝜙𝑟/2𝜋
𝑑𝑟 + 𝑗2𝜋(𝑓 − 𝑓𝑟)+ 𝜂(𝑓)
Frequency domain Model
HSVD based water suppression
𝑆 = 𝑊𝐻𝑇
Spectra from Voxels
Spectra of sources
Source abundancies in the grid
3Aim: Suppress the large water peak from all the voxels
Löwner based water suppression- Löwner matrix
• For a function 𝑆(𝑡) evaluated at 𝑇 = 𝑡1, 𝑡2, … . , 𝑡𝑁 . Partition T into two disjoint point sets 𝑋 = 𝑥1, 𝑥2, … . , 𝑥𝐼and 𝑦 = 𝑦1, 𝑦2, … . , 𝑦𝐽 , then Löwner matrix is given by:
• A Löwner matrix constructed by a rational function of degree-R will have a rank-R.
• The BSS problem 𝑆 = 𝑊𝐻𝑇 can be formulated using Löwner matrix/tensor*.
𝐿 =
𝑆 𝑥1 − 𝑆 𝑦1𝑥1 − 𝑦1
⋯𝑆 𝑥1 − 𝑆 𝑦𝐽
𝑥1 − 𝑦𝐽⋮ ⋱ ⋮
𝑆 𝑥𝐼 − 𝑆 𝑦1𝑥𝐼 − 𝑦1
⋯𝑆 𝑥𝐼 − 𝑆 𝑦𝐽
𝑥𝐼 − 𝑦𝐽
L𝑠 = 𝑟=1
𝑅
𝐿𝑊𝑟 ⊗ ℎ𝑟
4*O. Debals, M. Van Barel, and L. De Lathauwer, “Löwner-Based Blind Signal Separation of Rational Functions With Applications,” Signal Processing, IEEE Transactions on, vol. 64, no. 8, pp. 1909–1918, April 2016.
Löwner based water suppression*- CPD
• For each voxel in the MRSI signal construct a Löwnermatrix from the spectra and stack then to form a tensor.
• Each individual component can be well approximated by a degree-1 rational function, BSS reduces to CPD.
L𝑠 ≈ 𝑟=1
𝑅
𝑎𝑟 ⊗ 𝑏𝑟 ⊗ ℎ𝑟
5*H. N. Bharath, O. Debals, D. M. Sima, U. Himmelreich, L. De Lathauwer, S. Van Huffel, “Löwner Based Method for Residual Water Suppression in 1H Magnetic Resonance Spectroscopic Imaging ”, Submitted to IEEE transactions on biomedical engineering.
Löwner based water suppression – Method
• Estimate the Rational function parameters from mode-1 and mode-2 factor matrices using Least squares.
• Extend the sources outside the region of interest using the estimated parameters.
• Calculate the abundancies hk from extended sources and measured spectra using least squares.
• Water component is estimated using only the sources and amplitudes that are in the water frequency range (4.2-6 ppm).
• Finally the water component is suppressed by subtracting the estimated signal from the measured signal.
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Löwner based water suppression- Baseline• Problem: In some voxels, water suppression will result in a
baseline at the edges of the spectra.
• Model the baseline using polynomial function by adding it to the source matrix 𝑊.
𝑊𝑝𝑜𝑙𝑦 =𝑤11 ⋯ 𝑤1𝑅⋮ ⋱ ⋮
𝑤𝑁1 ⋯ 𝑤𝑁𝑅
1 ⋯ 𝑓1𝑑⋮ ⋱ ⋮1 ⋯ 𝑓𝑁𝑑
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Löwner based water suppression- Results
Box-plot of error on simulated MRSI data Box-plot of difference in variance on in-vivo MRSI data
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APPLICATION 2: Tissue type differentiationof Gliomas
Grade IV Glioblastoma patient:Edema
Active tumor
Necrosis
� Gliomas: 30% of all primary brain tumors and 80% of the malignant brain tumors.� WHO grade of malignancy: grade I-IV.� 5-year survival rates:
�Anaplastic astrocytoma(grade III): 26%�Glioblastoma multiforme (grade IV): 5%
Aim: To identify active tumor and tumor core pathological region
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Tissue type differentiation of Gliomas
NAACho
Lip
ppm
Normal
Tumor
Necrosis
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Method 1: NCPD applied to MRSI*
• It reduces the length of spectra without losing vital information required for tumor tissue type differentiation.
11*H. N. Bharath, D. M. Sima, N. Sauwen, U. Himmelreich, L. D. Lathauwer, and S. V. Huffel, “Non-negative canonical polyadic decomposition for tissue type differentiation in gliomas,” IEEE Journal of Biomedical and Health Informatics, vol. PP, no. 99, pp. 1–1, 2016
Method 1: NCPD applied to MRSI- XXT tensor
• Construct a 3-D tensor by stacking XXT from each voxel.• It gives more weight to the peaks and makes the signal
smoother. • MRSI tensor couples the peaks in the spectra because of
the XXT in the frontal slices.
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Method 1: NCPD applied to MRSI- NCPD
• Non-negative constraint is applied on all 3-modes.• To maintain symmetry in frontal slices common factor (S) is
used in both mode 1 and mode 2.
𝑇 ≈ [𝑆, 𝑆, 𝐻] = 𝑟=1
𝐾
𝑆 : , 𝑟 о 𝑆 : , 𝑟 о 𝐻(: , 𝑟)
• Non-negative CPD is performed in Tensorlab* toolboxusing structured data fusion.
13* Vervliet N., Debals O., Sorber L., Van Barel M. and De Lathauwer L. Tensorlab 3.0, Available online, Mar. 2016. URL: http://www.tensorlab.net/
Method 1: NCPD applied to MRSI:- NCPD-l1• Here, we assume that spectra corresponding to each voxel
belong to a particular tissue type, therefore the factor matrix H will be sparse.
• Non-negative CPD with l1 regularization on the abundances H.
[𝑆∗, 𝐻∗] = min𝑆,𝐻
𝑇 − 𝑟=1
𝐾
𝑆 : , 𝑟 о 𝑆 : , 𝑟 о 𝐻(: , 𝑟),
2
2
+𝜆 𝑉𝑒𝑐 𝐻 1
Where λ controls the sparsity in H. • Use more sources (higher rank) to accommodate for
artifacts and variations within tissue types.• Source Spectra are recovered from least squares:
𝑆 = (𝐻†𝑌𝑇)𝑇
𝐻† is the pseudo inverse of H obtained from Non-negativeCPD.
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Method 1: NCPD applied to MRSI- Results
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Method 1: NCPD applied to MRSI- Results
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Method 2: NCPD applied to multiparametric MRI*Conventional MRI PWI
DWI MRSI
17*H. N. Bharath, N. Sauwen, D. M. Sima, U. Himmelreich, L. De Lathauwer and S. Van Huffel, "Canonical polyadicdecomposition for tissue type differentiation using multi-parametric MRI in high-grade gliomas," 2016 24th European Signal Processing Conference (EUSIPCO), Budapest, 2016, pp. 547-551.
Method 2: NCPD applied to MP-MRI:-XXT tensor
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Method 2: NCPD applied to MP-MRI:- CPD
• To maintain symmetry in frontal slices common factor (S) isused in both mode-1 and mode-2.
• Non-negative constraint is applied on mode-3, H.• Also, l1 regularization in applied on the abundances H.
• Solved using structured data fusion method in Tensorlab*.
𝑆∗, 𝐻∗ = arg min𝑆,𝐻≥0
T − 𝑖=1
𝑅
)𝑆 : , 𝑖 𝑜 𝑆 : , 𝑖 𝑜 𝐻(: , 𝑖
2
2
+ 𝜆 )𝑉𝑒𝑐(𝐻 1
19* Vervliet N., Debals O., Sorber L., Van Barel M. and De Lathauwer L. Tensorlab 3.0, Available online, Mar. 2016. URL: http://www.tensorlab.net/
Method 2: NCPD applied to MP-MRI:- Results
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Method 2: NCPD applied to MP-MRI:- results
Constrained CPD-l1 hNMFDice
TumorDiceCore
Tumor sourceCorrelation
DiceTumor
DiceCore
Tumor sourceCorrelation
Mean 0.83 0.87 0.95 0.78 0.85 0.81Standard deviation 0.07 0.1 0.05 0.09 0.13 0.19
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